Riemann surfaces, lecture 6 M. Verbitsky Riemann surfaces lecture 6: hyperbolic plane Misha Verbitsky Universit´ e Libre de Bruxelles November 10, 2015 1
Riemann surfaces, lecture 6 M. Verbitsky Riemannian manifolds (reminder) DEFINITION: Let h ∈ Sym 2 T ∗ M be a symmetric 2-form on a manifold which satisfies h ( x, x ) > 0 for any non-zero tangent vector x . Then h is called Riemannian metric , of Riemannian structure , and ( M, h ) Riemannian manifold . DEFINITION: For any x.y ∈ M , and any path γ : [ a, b ] − → M connecting γ | dγ dt | dt , where | dγ x and y , consider the length of γ defined as L ( γ ) = � dt | = h ( dγ dt , dγ dt ) 1 / 2 . Define the geodesic distance as d ( x, y ) = inf γ L ( γ ), where infimum is taken for all paths connecting x and y . EXERCISE: Prove that the geodesic distance satisfies triangle inequality and defines metric on M . EXERCISE: Prove that this metric induces the standard topology on M . EXAMPLE: Let M = R n , h = � i dx 2 i . Prove that the geodesic distance coincides with d ( x, y ) = | x − y | . EXERCISE: Using partition of unity, prove that any manifold admits a Riemannian structure. 2
Riemann surfaces, lecture 6 M. Verbitsky Hermitian structures (reminder) DEFINITION: A Riemannia metric h on an almost complex manifold is called Hermitian if h ( x, y ) = h ( Ix, Iy ). REMARK: Given any Riemannian metric g on an almost complex manifold, a Hermitian metric h can be obtained as h = g + I ( g ) , where I ( g )( x, y ) = g ( I ( x ) , I ( y )) . REMARK: Let I be a complex structure operator on a real vector space V , and g – a Hermitian metric. Then the bilinear form ω ( x, y ) := g ( x, Iy ) Indeed, ω ( x, y ) = g ( x, Iy ) = g ( Ix, I 2 y ) = − g ( Ix, y ) = is skew-symmetric. − ω ( y, x ). DEFINITION: A skew-symmetric form ω ( x, y ) is called an Hermitian form on ( V, I ). REMARK: In the triple I, g, ω , each element can recovered from the other two. 3
Riemann surfaces, lecture 6 M. Verbitsky Conformal structure (reminder) DEFINITION: Let h, h ′ be Riemannian structures on M . These Riemannian structures are called conformally equivalent if h ′ = fh , where f is a positive smooth function. DEFINITION: Conformal structure on M is a class of conformal equiva- lence of Riemannian metrics. CLAIM: Let I be an almost complex structure on a 2-dimensional Riemannian manifold, and h, h ′ two Hermitian metrics. Then h and h ′ are conformally equivalent . Conversely, any metric conformally equivalent to Hermitian is Hermitian. REMARK: The last statement is clear from the definition, and true in any dimension. 4
Riemann surfaces, lecture 6 M. Verbitsky Conformal structures and almost complex structures (reminder) REMARK: The following theorem implies that almost complex structures on a 2-dimensional oriented manifold are equivalent to conformal structures. THEOREM: Let M be a 2-dimensional oriented manifold. Given a complex structure I , let ν be the conformal class of its Hermitian metric. Then ν is determined by I , and it determines I uniquely. DEFINITION: A Riemann surface is a complex manifold of dimension 1, or (equivalently) an oriented 2-manifold equipped with a conformal structure. A map from one Riemann surface to another is holomorphic if and only if it preserves the conformal structure. 5
Riemann surfaces, lecture 6 M. Verbitsky Homogeneous spaces (reminder) DEFINITION: A Lie group is a smooth manifold equipped with a group structure such that the group operations are smooth. Lie group G acts on a manifold M if the group action is given by the smooth map G × M − → M . DEFINITION: Let G be a Lie group acting on a manifold M transitively. Then M is called a homogeneous space . For any x ∈ M the subgroup St x ( G ) = { g ∈ G | g ( x ) = x } is called stabilizer of a point x , or isotropy subgroup . CLAIM: For any homogeneous manifold M with transitive action of G , one has M = G/H , where H = St x ( G ) is an isotropy subgroup. Proof: The natural surjective map G − → M putting g to g ( x ) identifies M with the space of conjugacy classes G/H . REMARK: Let g ( x ) = y . Then St x ( G ) g = St y ( G ): all the isotropy groups are conjugate. 6
Riemann surfaces, lecture 6 M. Verbitsky Isotropy representation (reminder) DEFINITION: Let M = G/H be a homogeneous space, x ∈ M and St x ( G ) the corresponding stabilizer group. The isotropy representation is the nat- ural action of St x ( G ) on T x M . DEFINITION: A tensor Φ on a homogeneous manifold M = G/H is called invariant if it is mapped to itself by all diffeomorphisms which come from g ∈ G . REMARK: Let Φ x be an isotropy invariant tensor on St x ( G ). For any y ∈ M obtained as y = g ( x ), consider the tensor Φ y on T y M obtained as Φ y := g (Φ). The choice of g is not unique, however, for another g ′ ∈ G which satisfies g ′ ( x ) = y , we have g = g ′ h where h ∈ St x ( G ). Since Φ is h -invariant, the tensor Φ y is independent from the choice of g . We proved THEOREM: Homogeneous tensors on M = G/H are in bijective cor- respondence with isotropy invariant tensors on T x M , for any x ∈ M . 7
Riemann surfaces, lecture 6 M. Verbitsky Space forms (reminder) DEFINITION: Simply connected space form is a homogeneous manifold of one of the following types: positive curvature: S n (an n -dimensional sphere), equipped with an action of the group SO ( n + 1) of rotations zero curvature: R n (an n -dimensional Euclidean space), equipped with an action of isometries negative curvature: SO (1 , n ) /SO ( n ), equipped with the natural SO (1 , n )- action. This space is also called hyperbolic space , and in dimension 2 hy- perbolic plane or Poincar´ e plane or Bolyai-Lobachevsky plane 8
Riemann surfaces, lecture 6 M. Verbitsky Riemannian metric on space forms (reminder) LEMMA: Let G = SO ( n ) act on R n in a natural way. Then there exists a unique G -invariant symmetric 2-form: the standard Euclidean metric. Proof: Let g, g ′ be two G -invariant symmetric 2-forms. Since S n − 1 is an Multiplying g ′ by orbit of G , we have g ( x, x ) = g ( y, y ) for any x, y ∈ S n − 1 . a constant, we may assume that g ( x, x ) = g ′ ( x, x ) for any x ∈ S n − 1 . Then g ( λx, λx ) = g ′ ( λx, λx ) for any x ∈ S n − 1 , λ ∈ R ; however, all vectors can be written as λx . COROLLARY: Let M = G/H be a simply connected space form. Then M admits a unique, up to a constant multiplier, G -invariant Riemannian form. Proof: The isotropy group is SO ( n − 1) in all three cases, and the previous lemma can be applied. REMARK: From now on, all space forms are assumed to be homoge- neous Riemannian manifolds . 9
Riemann surfaces, lecture 6 M. Verbitsky Some low-dimensional Lie group isomorphisms (reminder) DEFINITION: Lie algebra of a Lie group G is the Lie algebra Lie( G ) of left- invariant vector fields. Adjoint representation of G is the standard action of G on Lie( G ). For a Lie group G = GL ( n ), SL ( n ), etc., PGL ( n ), PSL ( n ), etc. denote the image of G in GL (Lie( G )) with respect to the adjoint action. REMARK: This is the same as a quotient G/Z by the centre of G . DEFINITION: Define SO (1 , 2) as the group of orthogonal matrices on a 3-dimensional space equipped with a scalar product of signature (1,2), and → C 2 preserving a pseudio- U (1 , 1) as the group of complex linear maps C 2 − Hermitian form of signature (1,1). THEOREM: The groups PU (1 , 1) , PSL (2 , R ) and SO (1 , 2) are isomor- phic. Proof: Isomorphism PU (1 , 1) = SO (1 , 2) will be established later in this lec- ture. To see PSL (2 , R ) ∼ = SO (1 , 2), consider the Killing form κ on the Lie algebra sl (2 , R ), a, b − → Tr( ab ). Check that it has signature (1 , 2) . Then the image of SL (2 , R ) in automorphisms of its Lie algebra is mapped to SO ( sl (2 , R ) , κ ) = SO (1 , 2) . Both groups are 3-dimensional, hence it is an isomorphism. 10
Riemann surfaces, lecture 6 M. Verbitsky Poincar´ e-Koebe uniformization theorem (reminder) DEFINITION: A Riemannian manifold of constant curvature is a Rie- mannian manifold which is locally isometric to a space form. THEOREM: (Poincar´ e-Koebe uniformization theorem) Let M be a Rie- mann surface. Then M admits a unique complete metric of constant curvature in the same conformal class. COROLLARY: Any Riemann surface is a quotient of a space form X by a discrete group of isometries Γ ⊂ Iso( X ) . COROLLARY: Any simply connected Riemann surface is conformally equivalent to a space form. REMARK: We shall prove some cases of the uniformization theorem in later lectures. Today’s subject: classify conformal automorphisms of all space forms. 11
Riemann surfaces, lecture 6 M. Verbitsky Laurent power series THEOREM: (Laurent theorem) Let f be a holomorphic function on an annulus (that is, a ring) R = { z | α < | z | < β } . i ∈ Z z i a i Then f can be expressed as a Laurent power series f ( z ) = � converging in R . Proof: Same as Cauchy formula. REMARK: This theorem remains valid if α = 0 and β = ∞ . C ∗ − REMARK: A function ϕ : → C uniquely determines its Laurent power series. Indeed, residue of z k ϕ in 0 is √− 1 2 πa − k − 1 . C ∗ − i ∈ Z z i a i REMARK: Let ϕ : → C be a holomorphic function, and ϕ = � Then ψ ( z ) := ϕ ( z − 1 ) has Laurent polynomial its Laurent power series. i ∈ Z z − i a i . ψ = � 12
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